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Number T(n,k) of ordered pairs of partitions of n with exactly k common parts; triangle T(n,k), n>=0, 0<=k<=n, read by rows.
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%I #26 Feb 11 2024 18:56:04

%S 1,0,1,2,1,1,4,3,1,1,12,7,4,1,1,16,19,8,4,1,1,48,35,23,9,4,1,1,60,83,

%T 43,24,9,4,1,1,148,143,106,47,25,9,4,1,1,220,291,186,115,48,25,9,4,1,

%U 1,438,511,397,210,119,49,25,9,4,1,1,618,949,697,444,219,120,49,25,9,4,1,1

%N Number T(n,k) of ordered pairs of partitions of n with exactly k common parts; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

%H Alois P. Heinz, <a href="/A370005/b370005.txt">Rows n = 0..200, flattened</a>

%e T(4,0) = 12: (1111,22), (1111,4), (211,4), (22,1111), (22,31), (22,4), (31,22), (31,4), (4,1111), (4,211), (4,22), (4,31).

%e T(4,1) = 7: (1111,31), (211,22), (211,31), (22,211), (31,1111), (31,211), (4,4).

%e T(4,2) = 4: (1111,211), (211,1111), (22,22), (31,31).

%e T(4,3) = 1: (211,211).

%e T(4,4) = 1: (1111,1111).

%e Triangle T(n,k) begins:

%e 1;

%e 0, 1;

%e 2, 1, 1;

%e 4, 3, 1, 1;

%e 12, 7, 4, 1, 1;

%e 16, 19, 8, 4, 1, 1;

%e 48, 35, 23, 9, 4, 1, 1;

%e 60, 83, 43, 24, 9, 4, 1, 1;

%e 148, 143, 106, 47, 25, 9, 4, 1, 1;

%e 220, 291, 186, 115, 48, 25, 9, 4, 1, 1;

%e 438, 511, 397, 210, 119, 49, 25, 9, 4, 1, 1;

%e ...

%p b:= proc(n, m, i) option remember; `if`(m=0, 1, `if`(i<1, 0,

%p add(add(expand(b(sort([n-i*j, m-i*h])[], i-1)*

%p x^min(j, h)), h=0..m/i), j=0..n/i)))

%p end:

%p T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n$3)):

%p seq(T(n), n=0..12);

%Y Column k=0 gives A054440.

%Y Row sums and T(2n,n) give A001255.

%Y Cf. A000041, A000290, A260669, A370207.

%K nonn,tabl

%O 0,4

%A _Alois P. Heinz_, Feb 07 2024